Mathematical Methods in the Applied Sciences Math. Meth. Appl. Sci., 22, 1189}1200 (1999) MOS subject classi"cation: 78 A 25; 35 Q 60 Asymptotic Stability for a Non-local Problem in Electromagnetism Carlo Alberto Bosello* Dip. di Matematica, Universita ` di Bologna, P.zza Porta San Donato 5, 40127 Bologna, Italy Communicated by D. S. Jones In this paper, constitutive equations of non-local type are coupled with Maxwell equations and the resulting di!erential problem is studied. A weak formulation is given for an initial}boundary-value problem for Maxwell equations in a medium obeying such constitutive equations with perfectly conducting boundary, and it is shown that such a problem admits at most one solution. The uniqueness theorem is then shown to imply the density of the range of a certain operator in the space of solutions and this result, together with an a priori energy inequality, is used to prove existence of solutions. Then the study of asymptotic stability of solutions is addressed. In particular, solutions are shown to be ¸ in time over (0, R). Finally, a brief description is given of the alternative problem arising when more general constitutive equations are used. Copyright  1999 John Wiley & Sons, Ltd. 1. Introduction Non-local properties for electromagnetic materials have been studied in [3, 2]. Such media take into account a quadrupole internal structure [5, 6], and are described through constitutive equations such as D(x, t)"  E (x, t)!(   ) E (x, t))! ) (  E (x, t)). (1) However, arguments of a physical nature which we shall discuss in section 8 lead us to consider the constitutive equations: D(x, t)"  E (x, t)#(E(x, t)), (2) B (x, t)"  H(x, t), (3) J (x, t)"E (x, t), (4) where the vectors D, E, B, H and J are, respectively, the electric displacement, the electric "eld, the magnetic induction, the magnetic "eld and the current density. Note * Correspondence to: C. A. Bosello, Dip. di Matematica, Universita` di Bologna, P.zza Porta San Donato 5, 40127 Bologna, Italy. CCC 0170}4214/99/141189}12 $17.50 Received 3 February 1999 Copyright  1999 John Wiley & Sons, Ltd.